Continuous-variable entanglement in a two-mode lossy cavity: an exact solution
CContinuous-variable entanglement in a two-mode lossy cavity: an exact solution
Colin Vendromin ∗ and Marc M. Dignam Department of Physics, Engineering Physics, and Astronomy,Queen’s University, Kingston, Ontario K7L 3N6, Canada (Dated: December 4, 2020)Continuous-variable (CV) entanglement is a valuable resource in the field of quantum information.One source of CV entanglement is the correlations between the position and momentum of photons ina two-mode squeezed state of light. In this paper, we theoretically study the generation of squeezedstates, via spontaneous parametric downconversion (SPDC), inside a two-mode lossy cavity thatis pumped with a classical optical pulse. The dynamics of the density operator in the cavity ismodelled using the Lindblad master equation, and we show that the exact solution to this modelis the density operator for a two-mode squeezed thermal state, with a time-dependent squeezingamplitude and average thermal photon number for each mode. We derive an expression for themaximum entanglement inside the cavity that depends crucially on the difference in the lossesbetween the two modes. We apply our exact solution to the important example of a microringresonator that is pumped with a Gaussian pulse. The expressions that we derive will help researchersoptimize CV entanglement in lossy cavities.
I. INTRODUCTION
Entanglement serves as a basis for many applicationsin the field of quantum information [1], such as quan-tum teleportation [2, 3] and quantum key distribution[4, 5]. Entanglement can occur as correlations betweendiscrete variables of two particles, such as the polar-ization of two photons, or as the correlations betweencontinuous variables, such as position and momentumof photons. In quantum-optical approaches to quantumcomputing, those that use discrete-variable (DV) entan-glement can achieve high-fidelity operations, but theiroperation is probabilistic. The use of continuous-variable(CV) entanglement has the advantage that it can achievedeterministic operations, but is not able to achieve ashigh-fidelity operations [6]. To remedy this trade-off andattempt to achieve large-scale quantum computing, hy-brid approaches that utilize both CV and DV have beendemonstrated [7].In the context of quantum optics, two-mode squeezedstates are routinely used as a source of CV entangle-ment. They can be generated by mixing two single-modesqueezed states on a beam-splitter [6], or via a nonlinearinteraction [8] such as spontaneous parametric downcon-version (SPDC), where a strong coherent pump field in-teracts with a material that has a χ (2) nonlinearity. Theresulting two beams of squeezed light are entangled bythe correlations between the position and momentum ofthe photons in each beam.The amount of CV entanglement in the squeezed stateincreases with the amount of squeezing, so in the limit ofinfinite squeezing the two-mode squeezed state resemblesa maximally-entangled EPR state [9]. To enhance theamount of squeezing, and thus entanglement, it is bene-ficial for the SPDC interaction to occur in a cavity that ∗ [email protected] is resonant with the pump field, as well as the generatedsignal and idler fields.In this work, we model the evolution of a squeezedstate inside a lossy multimode cavity. We model the evo-lution with the Lindblad master equation. We show thatthe exact solution to this model is a two-mode squeezedthermal state for all time, with time-dependent squeezingamplitude, and thermal photon number for each mode.In previous work [10] we studied single-mode squeezedstates in a lossy cavity. We modelled the dynamics of thegenerated squeezed light using the Lindblad master equa-tion for a single-mode lossy cavity and derived the exactsolution for the state in the cavity to be a squeezed ther-mal state. This present work builds on our previous workby including two modes that have different frequenciesand loss rates. As one of the main results in this paper,we derive an analytic expression for the entanglement inthe cavity as a function of the time-dependent squeez-ing and mode thermal photon numbers. This expres-sion includes an explicit dependence on the mode lossesand shows that the entanglement is maximized when thelosses of the two modes are equal.An important structure for generating CV entangle-ment is a microring resonator side-coupled to a waveg-uide. It enhances the nonlinear interaction that producessqueezed light and can be integrated on a chip [11–14].In the second part of the paper, we apply our model to aside-coupled ring resonator that is pumped with a Gaus-sian pulse that is coupled in from the linear waveguide.We derive a semi-analytic expression for the maximumentanglement in the ring that depends on the pump du-ration, coupling and scattering loss in the ring.The paper is organized as follows. In Sec. II we de-rive the two-mode squeezed thermal state solution, andgive a set of coupled first-order differential equations forthe squeezing amplitude and thermal photon numbers ofeach mode. Using our exact solution for the state in thecavity, in Sec. III we derive an expression for the insepa-rability of the state, and the maximum entanglement in a r X i v : . [ qu a n t - ph ] D ec the cavity as a function of the input pump amplitude. InSec. IV we apply our theory to the example of generatingCV entanglement in a side-coupled microring resonatorpumped with a Gaussian input pulse. Finally in Sec. Vwe present our conclusions. II. THE LINDBLAD MASTER EQUATION FORA TWO-MODE LOSSY CAVITY
In this section we derive the exact solution to the Lind-blad master equation for the generation of a two-modesqueezed state in a lossy cavity. The cavity is pumpedwith a strong pump field, and signal and idler photonpairs are generated via spontaneous parametric downcon-version (SPDC) with frequencies ω and ω , respectively.We assume that the cavity is resonant at the frequenciesof the signal and idler. Treating the pump as a strongclassical field and neglecting depletion of the pump due tothe nonlinear interaction, the Hamiltonian for the signaland idler photons in the cavity is given by [15]ˆ H = (cid:126) ω ˆ b † ˆ b + (cid:126) ω ˆ b † ˆ b + γ ∗ E ∗ P ( t )ˆ b ˆ b + γ E P ( t )ˆ b † ˆ b † , (1)where, ˆ b (ˆ b † ) and ˆ b (ˆ b † ) destroy (create) photons in thecavity in the mode 1 and 2 with frequency ω and ω ,respectively. The last two terms in Eq. (1) account forthe nonlinear SPDC interaction that generates the two-mode squeezed light, where the second-order nonlinearcoefficient is γ = − i (cid:126) χ (2) √ ω ω , where χ (2) is an effec-tive second-order nonlinear susceptibility. Here E P ( t ) isthe positive frequency part of the time-dependent classi-cal electric field, E P ( t ), where E P ( t ) = E P ( t ) + E ∗ P ( t ). Inall that follows, only the positive frequency part of theelectric field is used, because we make the rotating waveapproximation.To gain some insight into the state of the light in-side the cavity, first consider the case that the cav-ity is lossless and the pump is a continuous wave with E P ( t ) = E exp( − i ( ω + ω ) t ). Then the state evolvesin time with a unitary two-mode squeezing operator [15]given by ˆ S ( ξ ) = e ξ ∗ ˆ b ˆ b − ξ ˆ b † ˆ b † , (2)with a time-dependent complex squeezing parameter ξ = u exp( iφ ), related to the pump amplitude by ξ = itγ E / (cid:126) .Now we consider the general case where the cavity hasloss and there is a pulsed pump. In this case, the dynam-ics of the density operator ˆ ρ in the cavity are modelledusing the Lindblad master equation [16]: d ˆ ρdt = − i (cid:126) (cid:104) ˆ H, ˆ ρ (cid:105) + 12 (cid:88) j =1 Γ j (cid:16) b j ˆ ρ ˆ b † j − ˆ b † j ˆ b j ˆ ρ − ˆ ρ ˆ b † j ˆ b j (cid:17) , (3)where the loss of the cavity is captured by intensity decayrates Γ and Γ for modes 1 and 2, respectively. With the inclusion of loss, the state does not simply evolve intime by operating with a squeezing operator.The main result of this paper is the derivation of theexact solution to the system of Eqs. (1) and (3). First,however, we consider a simpler situation where there is nopump in the cavity ( E P ( t ) = 0), so that the last two termsin the Hamiltonian in Eq. (1) are zero. In this case, weassume that the initial state is a two-mode thermal state,which is formed by the product of single-mode thermalstates: ˆ ρ th = (cid:89) j =1 (1 − x j ) ( x j ) ˆ n j , (4)where x j = exp (cid:18) − (cid:126) ω j k B T j (cid:19) (5)and ˆ n j is the photon number operator for the mode j .Here, k B is the Boltzmann constant and T j is the effectivetemperature of mode j . The average number of photonsin the thermal state in each mode n j is given by n j = x j − x j . (6)We now prove that the state in the cavity remains a two-mode thermal state for all time, with an average thermalphoton number that decays exponentially over time. Thismeans that the solution to Eq. (3) when there is nointeraction term in the Hamiltonian in Eq. (1) can bewritten as ˆ ρ ( t ) = ˆ ρ th ( t ) , (7)where ˆ ρ th ( t ) is given in Eq. (4), except now the vari-able x j is time-dependent due to the dynamical thermalphoton number. Rearranging Eq. (7) we obtainˆ = [ˆ ρ th ( t )] − / ˆ ρ ( t ) [ˆ ρ th ( t )] − / , (8)where ˆ is the identity operator and we have used thefact that the thermal state operator is unitary. Takingthe time derivative of both sides of Eq. (8) yields0 = ddt (cid:16) ˆ ρ − / th ˆ ρ ˆ ρ − / th (cid:17) . (9)Applying the chain rule, we obtain0 = d ˆ ρ − / th dt ˆ ρ ˆ ρ − / th + ˆ ρ − / th d ˆ ρdt ˆ ρ − / th + ˆ ρ − / th ˆ ρ d ˆ ρ − / th dt . (10)Simplifying Eq. (10) using the derivative of ˆ ρ in Eq. (3)and the identity in Eq. (8), we eliminate ˆ ρ from all theterms in Eq. (10), and are left with terms that onlycontain ˆ ρ th and its derivative. Then, using the thermalstate in Eq. (4), we obtain0 = (cid:88) j =1 (cid:18) ˆ n j + ˆ x j x j − (cid:19) D j , (11)where D j = − x j dx j dt + Γ j ( x j − . (12)In order for Eq. (11) to be true for all times we musthave that D j = 0, which has the solution x j ( t ) = 11 + C e Γ j t , (13)where C is a constant determined by the initial condi-tions. Using Eqs. (5) and (6), we can rewrite Eq. (13)as n j ( t ) = n j (0)e − Γ j t , (14)which says that the average photon number in each modedecays from an initial value n j (0) at the rate Γ j . Thus,if we start with a thermal state, it will remain a ther-mal state for all times, but with different time-dependenttemperatures for the two modes.Our focus now is to include the pump, so that the lasttwo terms in the Hamiltonian in Eq. (1) are not zero.In this case, as we shall see, the two-mode thermal statewill be squeezed with the two-mode squeezing operatorˆ S in Eq. (2). Therefore, we propose that the solution toEq. (3) is the squeezed two-mode thermal stateˆ ρ ( t ) = ˆ S ( ξ ( t ))ˆ ρ th ( t ) ˆ S † ( ξ ( t )) , (15)where now the thermal photon number and squeezingparameter are time-dependent. Rearranging Eq. (15) weobtainˆ = [ˆ ρ th ( t )] − / ˆ S † ( ξ ( t ))ˆ ρ ˆ S ( ξ ( t )) [ˆ ρ th ( t )] − / . (16)Taking the time derivative of both sides, we obtain0 = ddt (cid:16) ˆ ρ − / th ˆ S † ˆ ρ ˆ S ˆ ρ − / th (cid:17) . (17)Applying the chain rule gives0 = d ˆ ρ − / th dt ˆ S † ˆ ρ ˆ S ˆ ρ − / th + ˆ ρ − / th ˆ S † ˆ ρ ˆ S d ˆ ρ − / th dt ++ ˆ ρ − / th d ˆ S † dt ˆ ρ ˆ S ˆ ρ − / th + ˆ ρ − / th ˆ S † ˆ ρ d ˆ Sdt ˆ ρ − / th + (18)+ ˆ ρ − / th ˆ S † d ˆ ρdt ˆ S ˆ ρ − / th . Following a process similar to the one above in the caseof no pump, in Appendix A we simplify the terms in Eq. (18) and show that for the equality to be satisfied forall times, the thermal photon numbers ( n ( t ) and n ( t )),squeezing amplitude ( u ( t )), and squeezing phase ( φ ( t ))must be solutions of the following first-order coupled dif-ferential equations: dn dt = n (cid:0) Γ sinh u − Γ cosh u (cid:1) + Γ sinh u, (19) dn dt = n (cid:0) Γ sinh u − Γ cosh u (cid:1) + Γ sinh u, (20) dudt = i (cid:126) (cid:0) E P γ e − iφ − E ∗ P γ ∗ e iφ (cid:1) − sinh(2 u ) n + n + 1 × (cid:18) Γ + Γ − Γ n − n ] (cid:19) , (21) dφdt = − ( ω + ω ) + (cid:0) E P γ e − iφ + E ∗ P γ ∗ e iφ (cid:1) (cid:126) tanh(2 u ) , (22)where we have omitted the time-dependencies in all vari-ables and E P ( t ) for simplicity. Eqs. (19) to (22) de-scribe the dynamics of the squeezing of the state in thecavity. If we let the decay rates of the two modes beidentical, i.e. Γ = Γ , this set of equations reduces tothe single mode squeezing equations from our previouswork [17]. Note that Eq. (22) only depends on the sum-frequency frequency ω s = ω + ω . Thus, the dynamics ofthe squeezed state does not depend on the individual fre-quencies of the signal and idler photons, except implicitlythrough the effective nonlinearity, γ .In order to obtain a specific solution to this set of first-order differential equations, an initial state in the cavitymust be specified. In all that follows, we take the initialstate in the cavity to be the vacuum. Other initial statescan be chosen, but we chose the vacuum state because,as we will show, it simplifies the form of the equationsand corresponds to physically-realizable experiments.We define an initial time t i when the state in the cavityis vacuum, that is u ( t i ) = 0, and n j ( t i ) = 0, respectively.This causes the denominator of the second term in Eq.(22) to be zero, because tanh(2 u ( t i )) = 0. In order toeliminate the singularity, we choose the initial squeezingphase such that the numerator of the second term is zero E P ( t i ) γ e − iφ ( t i ) + E ∗ P ( t i ) γ ∗ e iφ ( t i ) = 0 , (23)where the initial pump in the cavity E P ( t i ) is arbitrarilysmall but not zero. If we let the pump and nonlinearparameter be written as a general complex number as E P ( t i ) γ = |E P ( t i ) || γ | e iθ − iω s t i , then the solution to Eq.(23) for the initial squeezing phase is φ ( t i ) = − π/ − θ + ω s t i [18]. Iterating Eqs. (21) and (22) forward one stepin time from the initial condition, it can be shown thatthe squeezing phase for all future times is given by φ ( t ) = − ω s t + φ ( t i )= − ω s ( t − t i ) − π θ. (24)Therefore if the initial state is vacuum, then the squeez-ing phase does not depend on the thermal photon num-bers or squeezing amplitude, and simply rotates aroundthe origin of phase space with the sum-frequency ω s .Using the expression in Eq. (24) for φ ( t ) in Eq. (21),the equation for the squeezing amplitude becomes, dudt = |E P ( t ) || γ | (cid:126) − sinh(2 u ) n + n + 1 × (cid:18) Γ + Γ − Γ n − n ] (cid:19) . (25)It is convenient to write Eqs. (19), (20), and (25) interms of the dimensionless parameter ˜ t = Γ + t , whereΓ ± = (Γ ± Γ ) /
2. Doing this, we obtain dud ˜ t = g (˜ t )2 − sinh(2 u ) n + n + 1 (1 + ζ [ n − n ]) , (26) dn d ˜ t = n (cid:0) [1 − ζ ] sinh u − [1 + ζ ] cosh u (cid:1) ++ [1 − ζ ] sinh u, (27) dn d ˜ t = n (cid:0) [1 + ζ ] sinh u − [1 − ζ ] cosh u (cid:1) ++ [1 + ζ ] sinh u, (28)where ζ = Γ − / Γ + is proportional to the difference of thetwo decay rates, and g ( t ) is the pumping strength in thering; it is defined as the rate of light generation in thecavity divided by the average decay rate of light out ofthe cavity g ( t ) ≡ |E P ( t ) || γ | (cid:126) Γ + . (29)Because the decay rates must be positive, it follows that | ζ | <
1. When ζ = 0, the decay rates of the modes areequal, the thermal photon numbers are the same, andconsequently the coupled equations reduce to the single-mode squeezing case.In summary, we have shown that the exact solutionto the Lindblad master equation for the generation ofa two-mode squeezed state in a lossy cavity is a two-mode squeezed thermal state and we have derived a setof coupled first-order differential equations that describethe dynamics of the state as a function of the pumpingstrength and cavity decay rates for each mode. We nowuse the two-mode squeezed thermal state to give a con-dition for the continuous variable entanglement in thecavity, and derive an expression for the maximum entan-glement. III. CONTINUOUS-VARIABLEENTANGLEMENT IN A TWO-MODESQUEEZED THERMAL STATE
In this section we give the entanglement condition forthe light in the cavity and derive a semi-analytic expres-sion for the maximum amount of entanglement as a func-tion of the pump and the thermal photon number differ-ence between the two modes. We define a general quadrature operatorˆ χ j ( β j ) = ˆ b j e iβ j + ˆ b † j e − iβ j , (30)where j = 1 and j = 2 are the two modes in the cavity,and β j is an angle in phase space. The quantum noise inthe quadrature is given by∆ χ j = (cid:113) tr (cid:0) ˆ ρ ˆ χ j (cid:1) − [tr (ˆ ρ ˆ χ j )] . (31)For the two-mode squeezed thermal state in Eq. (16),using the fact that tr(ˆ ρ ˆ χ j ) = 0, we obtain for the quadra-ture noise∆ χ j = (cid:114)
14 cosh(2 u ) + 12 cosh ( u ) n j + 12 sinh ( u ) n k , (32)where k = 1 if j = 2 and k = 2 if j = 1. The vacuumnoise in the quadrature is given by ∆ χ j = 1 /
2. Thereforethe quadrature noise is squeezed below the vacuum noisewhenever ∆ χ j < /
2. For the two-mode squeezed ther-mal state, the quadrature noise in Eq. (32) is representedas a circle in phase space with a radius always greaterthan or equal to 1 /
2, so there is no squeezing. Linearcombinations of the quadrature operators for mode 1 and2, however, can exhibit squeezing. To this end, we definetwo operators ˆ X and ˆ Y ˆ X ≡ ˆ χ ( β ) + ˆ χ ( β ) (33)ˆ Y ≡ ˆ χ (cid:16) β + π (cid:17) − ˆ χ (cid:16) β + π (cid:17) . (34)The operators ˆ X and ˆ Y are proportional to the sum anddifference of the individual position and momenta oper-ators of the light in each mode. The quadrature noise inˆ X and ˆ Y using the two-mode squeezed thermal state is (cid:10) (∆ X ) (cid:11) = 12 (1 + n + n ) × [cosh(2 u ) − cos ( φ + β + β ) sinh(2 u )] , (35)where φ is the time-dependent squeezing phase in Eq.(24). It is straightforward to show that (cid:10) (∆ Y ) (cid:11) = (cid:10) (∆ X ) (cid:11) . For certain phase relationships between β ( t ), β ( t ), and φ ( t ), the quadrature noise is reduced belowvacuum. Choosing the relationship between the quadra-ture phases be β ( t ) + β ( t ) = − φ ( t ), the fast oscillationsare removed and the quadrature noise is exponentiallysqueezed to give (cid:10) (∆ X ) (cid:11) = 12 [1 + n ( t ) + n ( t )] e − u ( t ) . (36)In the limit of infinite squeezing, u → ∞ , the uncer-tainty in the relative quadrature operator goes to zero, (cid:10) (∆ X ) (cid:11) →
0. Thus, a measurement on ˆ χ would give anexact prediction of ˆ χ , such that the two modes are per-fectly correlated. However, for finite squeezing the twomodes will not be perfectly correlated. To evaluate thedegree of entanglement between the two modes, we de-fine the correlation variance as the sum of the quadraturenoises in the relative position and momentum operators∆ , = (cid:10) (∆ X ) (cid:11) + (cid:10) (∆ Y ) (cid:11) , (37)and using Eq. (36) we obtain∆ , ( t ) = [1 + n ( t ) + n ( t )] e − u ( t ) . (38)It has been proved by Duan et al. [19] and Simon [20]that the two-mode squeezed state is inseparable, andtherefore entangled, when the correlation variance is lessthan 1, ∆ , ( t ) < . (39)This condition is achieved when the exponential squeez-ing factor (exp( − u )) overcomes the thermal noise factorin front.The correlation variance in Eq. (38) is time depen-dent because the squeezing amplitude and thermal pho-ton numbers are time dependent. If we assume that thepump is pulsed, then there should be a time when thecorrelation variance is minimized and the state exhibitsthe maximum amount of entanglement. In order to findan expression for the maximum entanglement we find theextreme point of the correlation variance in Eq. (38), i.e. we solve the following equation: d ∆ , dt (cid:12)(cid:12)(cid:12)(cid:12) t = t min = 0 . (40)Simplifying this equation by using Eqs. (26) - (28) weobtain, (cid:0) ∆ , (cid:1) min = 1 + ζ [ n ( t min ) − n ( t min )]1 + g ( t min ) . (41)Thus, even a small difference in the decay rates of thetwo modes can cause an increase in the minimum valueof the correlation variance and decrease the degree ofentanglement. In situations where we can treat | ζ | as asmall perturbation, we can neglect the second term inthe numerator of Eq. (41) and write, (cid:0) ∆ , (cid:1) min ≈
11 + g ( t min ) . (42)This represents an ideal case, when the two decay ratesfor the modes are the same and we can drop the ζ de-pendence. In this case, better entanglement is alwaysachieved by increasing the pumping strength. We do nothave an expression for the time when the correlation vari-ance is minimum, t min , so Eq. (41) and Eq. (42) muststill be evaluated numerically. However, if the pump fieldis a continuous wave, then the pumping strength g ( t )does not depend on time, and the pumping strength at t min in Eq. (42) can be replaced with essentially the am-plitude of the pump in the cavity. When the pump field FIG. 1: Schematic of the ring resonator side-coupled toa channel waveguide. The thick arrows represent thecomponents of the pump field (blue) with frequency ω P ,incident from the left, and the signal and idler fields(both red) with frequencies ω and ω , respectively.The cross-coupling coefficients and scattering lossparameters for the pump, signal, and idler fields are κ P , κ , and κ and a P , a , and a , respectively.is pulsed then, as we will show in the example below,it is often a good approximation to replace g ( t min ) withthe peak value of the pump in the cavity, g max . In thenext section, we will discuss the results of the entangle-ment for a two-mode squeezed thermal state generatedin a side-coupled ring resonator. IV. EXAMPLE: GENERATINGCONTINUOUS-VARIABLE ENTANGLEMENT INA RING RESONATOR
In this section we apply our formalism to a ring res-onator, which is optimized to produce a maximally-entangled state, and we study the dynamics of the en-tanglement. In previous work [17], we treated the sameproblem but for degenerate squeezing, where the signaland idler have the same frequency. In this example, wewill generalize our previous results to the case where thesignal and idler are distinguishable by their frequencyand loss.The ring resonator system we consider, shown in Fig.1, consists of a straight waveguide (channel) side-coupledto a ring waveguide with radius R . Both the ring andchannel are made from the same material that has a non-linear χ (2) response, but we ignore that nonlinear interac-tion in the channel, since the pump field is much weakerthere than in the ring. The classical pump field incidenton the ring is taken to be a classical Gaussian pulse. Thepositive frequency part of the pump field in the channeltakes the form E CH ( t ) = E (cid:114) T R τ exp (cid:18) − t τ (cid:19) exp( − iω P t ) , (43)where E is the amplitude, τ is the FWHM of the in-tensity of the Gaussian, ω P is the central frequency, and T R = 2 πRn eff /c is the ring round-trip time, where n eff is the effective index for the pump. We scale the am-plitude by 1 / √ τ so that the pulse energy is a constantthat is independent of its duration. The coupling of thepulse into the ring happens at a single coupling point, asindicated in Fig. 1. The pump field in the ring, E R ( ω ), iscalculated using a transfer matrix approach [17, 21], andis related to the field in the channel by E R ( ω ) = iκ P a P e iωT R − σ P a P e iωT R E CH ( ω ) , (44)where E ( ω ) is the Fourier transform of E ( t ) defined as E ( ω ) = (cid:90) ∞−∞ E ( t ) e iωt dt, (45)and the inverse Fourier transform as E ( t ) = 12 π (cid:90) ∞−∞ E ( ω ) e − iωt dω. (46)In Eq. (44), the constant κ P is the frequency-independent cross-coupling constant between the ringand channel, and is related to the self-coupling constant σ P through the lossless coupling relation: κ P + σ P = 1.The parameter a P is the amplitude attenuation for thepump in the ring after a single round-trip. It is re-lated to the power attenuation coefficient α P by a P =exp( − α P πR ). This is the power lost due to scatteringonly and not coupling. When a P = 1, there is no scat-tering loss.The buildup factor is defined as the ratio of the inten-sity in the ring to the channel and is given by B ( ω ) ≡ |E R ( ω ) | |E CH ( ω ) | = κ P a P − σ P a P cos( ωT R ) + σ P a P . (47)It contains resonant peaks when the incident light is onresonance with the ring, i.e. cos( ωT R ) = 1, and thebuildup is maximized. We choose the central frequencyof the pump pulse to be on resonance with the ring, ω P = 2 πm P /T R , where m P is a positive integer defin-ing the pump mode number. Inside the ring, the pumpfield generates signal and idler photons at the frequen-cies ω and ω , via SPDC, such that energy is conserved: ω P = ω + ω . The signal and idler fields are also res-onant with the ring, ω T R = 2 πm and ω T R = 2 πm ,where m and m are the mode numbers for the sig-nal and idler fields, respectively. For simplicity we as-sume perfect phase matching between the three fields,and that the effective index of refraction is the samefor the pump, signal, and idler modes. The latter as-sumption is experimentally realized in an AlN microringresonator with a waveguide width of 1 . µ m [12], andmakes the ring round-trip time the same for each field. Using these assumptions the mode numbers are relatedby m P = m + m .Generally, signal and idler photons will be generatedin a number of different pairs of ring modes, as long asthey satisfy energy conservation and phase matching. Tosimply our theory however, we assume that only the tworing modes m and m are perfectly phased matched, andsqueezed light generation in all other modes is neglected.Since the pump is pulsed, in general it will couple intomultiple ring modes depending on how wide its band-width is. To ensure that most of the pump light couplesinto a single mode m P we require that its duration islonger than the ring round-trip time ( τ > T R ). Doingso makes the pulse bandwidth overlap a single resonancepeak of the buildup factor and thus the adjacent peaksin the buildup factor do not significantly couple any lightinto the ring. The buildup factor for the central pumpfrequency ω P becomes, B ( ω P ) = κ P a P (1 − σ P a P ) . (48)In all that follows we are only interested in the limitthat the buildup factor is large, which only occurs when(1 − σ P a P ) (cid:28)
1, such that the denominator of Eq. (48)is small. This limit produces the largest entanglement inthe ring.Now we need to find an expression for the time-dependent pump field in the ring that we can use to solvefor the dynamics of the squeezed state. This is done bytaking the inverse Fourier transform of the field compo-nent in the ring. As we have shown in our previous work[17], in the limit when (1 − σ P a P ) (cid:28) E R ( t ) = τ κ P a P √ π e z ( t ) erfc[ z ( t )] (cid:112) T R E CH ( t ) , (49)where z ( t ) ≡ (1 − σ P a P ) τ (cid:112) T R − (cid:112) t τ . (50)Now to model the evolution of the state in the ring weuse the pump field in Eq. (49) in the coupled Eqs. (26)to (28). The pumping strength in the ring defined in Eq.(29) becomes g (˜ t ) = g κ P a P √ π e z (˜ t ) erfc[ z (˜ t )] (cid:112) (cid:115) ˜ τT R Γ + × exp (cid:18) − t ˜ τ (cid:19) , (51)where we define the dimensionless parameter g ≡ | γ |E (cid:126) Γ + , (52)which is proportional to the input pump pulse amplitude E divided by the average decay rate of the signal andidler mode Γ + , where we have used the same parame-ters defined above, ˜ t = Γ + t and ˜ τ = Γ + τ . At this pointwe could simply solve the dynamic Eqs. (26) to (28)numerically for any desired loss and pump parameters.However, as we shall show, a key parameter that affectsthe entanglement is the difference in the loss in the signaland idler modes, ζ . Therefore, to simplify the discussionof our results and to make the pumping strength in Eq.(51) dependent only on the pump coupling and loss pa-rameters, we assume that the average of the decay rate ofthe signal and idler is equal to the pump decay rate, Γ P , i.e. , Γ + ≡ Γ P , where when (1 − σ P a P ) (cid:28)
1, the pumpdecay rate is Γ P = 2(1 − σ P a P ) T R . (53)Then using Eq. (53) in Eq. (51), the pumping strengthin the ring can be written as g (˜ t ) = g κ P a P √ − σ P a P (cid:115) ˜ τ (cid:18) − t ˜ τ (cid:19) × √ π e z (˜ t ) erfc[ z (˜ t )] . (54)This completes the parameterization of our model.Our focus now is to examine the numerical results forthe correlation variance in Eq. (38) found by solving thedynamic equations with the pump field given by the ex-pression in Eq. (54). To do this, we use a 4th-orderRunge-Kutta method which has a run-time of about 1millisecond on a standard PC. Our model depends onthe parameters g , κ P , σ P , and a P , and the differencein the loss rates ζ . Thus, our results are effectively in-dependent of the ring radius, pump amplitude, and thenonlinear parameter. However, to make connection toa specific, realistic system, we choose a ring of radius R = 20 µ m made from AlGaAs, with an effective second-order nonlinear susceptibility of χ (2) = 11 pm / V [11].Additionally, we let the central frequency and amplitudeof the pump pulse be ω P = 2 π × . λ P = 775 nm)and E = 1 MV / cm. The frequencies of the signal andidler fields are ω = 2 π × .
68 THz ( λ = 1545 nm) and ω = 2 π × .
27 THz ( λ = 1555 nm), respectively. Withthese parameter choices, g = 4.In our previous work [17], we derived the optimumvalue for the pulse duration ˜ τ opt by taking the derivativeof the pumping strength in Eq. (54) with respect to τ ,at the peak value, and set it equal to zero. This resultsin the following accurate estimate of the optimum pulseduration of ˜ τ opt = 0 . (cid:112) . (55)The prefactor of 0.684 in this expression arises from thesolution of a transcendental equation, and this can beevaluated to arbitrary accuracy. The optimum value for σ P can be obtained [17] by taking the derivative of thepumping strength with respect to σ P , at the peak value and ˜ τ = ˜ τ opt , and set it equal to zero. For a loss param-eter of a P = 0 .
99, this results in an optimum value of σ P = 0 . ζ = 0. Since the photons in modes 1 and 2 arescattered from the ring at the same rate, the photon pairsare not separated due to scattering and the state in thering remains an entangled state with ∆ , < ζ = 1 /
3, which gives Γ = 2Γ and Γ = 2Γ P /
3. Thediscrepancy between the two decay rates means that pho-tons are being scattered from mode 1 twice the rate asfrom mode 2. Effectively, this means that half of the gen-erated entangled state is being traced-out by the scatter-ing; i.e. , photon 1 escapes the ring while photon 2 staysinside the ring. Thus, the state in the ring evolves intoa thermal state. This is why the correlation variance inFig. 2b goes above 1 after ˜ t ≈
2. After this time, thecorrelation variance becomes as large as 60 (not shown inplot) and the state in the ring is essentially a separabletwo-mode thermal state.In Fig. 3 the minimum of the correlation variance,∆ , ( t min ), is shown as a function of ˜ τ and ζ . The firstthing to notice is that it is symmetric about ζ = 0. Thisis because the product ζ ( n − n ) in Eq. (41) is alwayspositive. Next we observe that for a given τ , the bestentanglement is always where ζ = 0, because this makesthe numerator of Eq. (41) as small as possible ( i.e. , equalto 1). The global minimum in ∆ , ( t min ) (indicated bythe white cross) occurs with the τ that makes g ( t min ) aslarge as possible (see Eq. (42)), which occurs approxi-mately at ˜ τ opt (black line) given in Eq. (55). When weincrease or decrease the pulse duration from the optimumvalue, it causes ∆ , ( t min ) to increase. This is becausethe energy of our pulse is independent of its duration,so for long pulses its amplitude scales as 1 / √ τ and forshort pulses the energy is injected over too short a timeinterval for the intensity to buildup. In both cases, for along or short pulse, the pumping strength decreases andthere is less entanglement. When | ζ | >
0, the entangledstate is degraded by the unequal scattering of its photonpairs, and by increasing τ we observe a rapid decreaseof the entanglement. Conversely, when ζ = 0 and τ isincreased, then the decrease of the entanglement is lessrapid, because both photons are scattered at the samerate and the entanglement is better. However, when thepulse duration is shorter than the optimum, the entan-glement is not very sensitive to | ζ | .The thermal photon number in mode 1, evaluated at t min , is shown in Fig. 4 as a function ˜ τ and ζ for thesame parameters as above. A relatively large thermalphoton number implies that there is more generation ofsqueezed light, but if the thermal photon number be-comes too large, it destroys the entanglement (see Eq.(38)). There is an average thermal photon number ofapproximately 2 for each mode at the global minimumof the correlation variance. Where the average thermal (a)(b) FIG. 2: The time-dependent correlation variance for (a) ζ = 0, and (b) ζ = 1 /
3, pumping strength g = 4, using apulsed pump of duration ˜ τ opt , a pump scattering loss of a P = 0 .
99, and optimum self-coupling constant σ P = 0 . photon number is 10 there is more generation of squeezedlight, but the entanglement is weaker due to the increasedthermal noise. The thermal photon number in mode 2 isa mirror reflection, about ζ = 0, of mode 1, which canbe shown by letting ζ → − ζ in Eqs. (27) and (28).In both Fig. 2a and Fig. 2b, the minimum value of thecorrelation variance occurs very close to the time whenthe pumping strength is maximum in the ring ( g max ).Thus, in Eq. (42), which gives the approximate minimumvalue of the correlation variance, we can let g ( t min ) = g max . In our previous work [17] we showed that this isa valid approximation when the input pulse duration islonger than T R . Using Eq. (54) it can be shown thatthe pumping strength peaks at ˜ t = 1 for the optimumpulse duration. Putting ˜ t = 1 and ˜ τ opt into the pumpingstrength in Eq. (54) gives the following expression for FIG. 3: The minimum value of the correlation variance(∆ , ) min as a function of the difference between thelosses of the two-modes ζ and the pump pulse duration˜ τ . The optimum pulse duration ˜ τ opt given by Eq. (55)is shown with the black line, and the global minimum isindicated by the white cross. All other parameters arethe same as in Fig. (2).FIG. 4: The average number of thermal photons in mode 1,evaluated at the time when the correlation variance isminimum, n ( t min ), as a function of ζ and ˜ τ . All parametersare the same as in Fig. (3) g max : g max = 0 . g κ P a P √ − σ P a P . (56)Therefore, we can use Eq. (56) in Eq. (42) in place of g ( t min ) to obtain an expression for the minimum of thecorrelation variance when | ζ | (cid:28) g , κ P , σ P , and a P only, (cid:0) ∆ , (cid:1) min = (cid:20) . g κ P a P √ − σ P a P (cid:21) − . (57)FIG. 5: The global minimum in the correlation varianceas a function of g for pump scattering loss andoptimum self-coupling constant of a P = 0 .
99 and σ P = 0 .
868 (circle) and a P = 0 .
75 and σ P = 0 . σ P that minimizes Eq.(57) for a fixed g and a P is, σ P = 1 − (cid:112) − a P a P . (58)Putting this optimum coupling constant into Eq. (57)gives a good approximation to the global minimum inthe correlation variance for a given g and loss a P . InFig. 5, the global minimum in the correlation varianceis shown as a function of g for a P = 0 .
99 (circle) and a P = 0 .
75 (cross). The solid lines are the expression inEq. (57), and they show excellent agreement with the fullnumerical simulations across a wide range of g values forthe two scattering losses. Also shown is the sum of thethermal photon numbers in the two modes. This showsthat the cost associated with decreasing the correlationvariance is an increase in the overall thermal noise inthe ring. By increasing the scattering loss we observea decrease in the entanglement and the total thermalphoton number. In order to recover the entanglementwe can increase g , but this causes an increase in thethermal noise. For example, when a P = 0 .
99 and g = 4,the minimum of the correlation variance is about 0 . a P = 0 .
75 and keep g = 4, then the variance jumps to about 0 .
35 and thetotal thermal photon number decreases to about 2. Inorder to recover the variance, we would need to increaseto g = 7, and as a result we would increase the totalthermal photon number to about 20.The expression for the correlation variance in Eq. (38)is what one would infer from measurements in a homo- dyne detection experiment, if the local oscillators are per-fectly phase matched to the squeezing phase φ ( t ). It whatfollows, we will allow for a small angle deviation δθ fromperfect phase matching and see how this affects the cor-relation variance. The correlation variance when there isan angular deviation becomes∆ , = (1 + n + n ) [cosh(2 u ) − cos( δθ ) sinh(2 u )] . (59)If δθ = 0 we recover the perfect case in Eq. (38). InFig. 6a we show the minimum of the correlation vari-ance for an angular deviation of 5 mrad, with a P = 0 . σ P = 0 . g = 4. This angular deviation wasmeasured in a recent experiment [22]. Overall, the corre-lation variance is not changed significantly from the idealcase (when there is no offset), but there is a small increasein the global minimum from approximately 0.228 whenthere is no offset to approximately 0.229. In Fig. 6band Fig. 6c the angle offset is increased to δθ = 20 mradand δθ = 100 mrad, respectively, while keeping all otherparameters the same. For these two offsets, the globalminimum increases to approximately 0.235 and 0.282, re-spectively. The general trend created by increasing theangle offset is that the global minimum of the correlationvariance shifts to shorter pulse durations and the opti-mal region flattens out; becoming more sensitive to pulseduration and less sensitive to ζ . Thus, the discrepancybetween the two decay rates does not destroy the entan-glement as much when there is an angle offset. Instead,the pulse duration is much more destructive to the en-tanglement. When there is an angle offset, it is alwaysbetter to shorten the optimum pulse duration to improveentanglement. V. CONCLUSION
In conclusion, we have shown that the exact solu-tion to the Lindblad master equation for a two-modecavity pumped with a classical optical pulse is a two-mode squeezed thermal state for all time. We deriveda semi-analytic solution for the time-dependent squeez-ing amplitude, and thermal photon numbers for the twomodes, which we then used to derive an expression forthe maximum entanglement in the cavity (see Eq. (41)).The main parameter that affects the amount of entan-glement is the difference in the cavity losses between thetwo modes ζ . In the ideal case, the losses for the twomodes are the same ζ = 0 and the correlation variancejust scales inversely proportional to the peak pumpingstrength in the cavity (see Eq. (42)). However, the trade-off is that when the pumping strength is increased thetotal thermal noise also increases.We applied our theory to the important example ofgenerating CV entanglement in a ring resonator with dif-ferent losses for each mode. We derived a semi-analyticexpression for the maximum entanglement in the ring(see Eq. (57)) that depends only on the pump scattering0 (a) (b) (c) FIG. 6: The minimum in the correlation variance as a function fo ˜ τ and ζ for homodyning angles of (a) 5 mrad, (b)20 mrad, and (c) 100 mrad. All other parameters are the same as in Fig. (3).loss and coupling parameters. If the two modes have un-equal losses we demonstrated that it is always better todecrease the pulse duration from the optimum in order toachieve better entanglement. We considered the case ofa phase offset in a homodyne detection experiment thatwould cause a degradation of the measured entanglement,and we showed that the maximum entanglement could berecovered to a high degree by reducing the pulse durationfrom the optimum. Additionally, when the phase offsetis increased, it was shown that the entanglement is lesssensitive to the unequal losses of the two modes.These results will be of use to researchers that are try-ing to optimize CV entanglement in lossy cavities when the losses of each mode are different. In future work,we will apply this theory to the generation of two-modesqueezed light in a slab photonic crystal, where a three-mode cavity is side-coupled to a defect waveguide. ACKNOWLEDGEMENTS
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0, which results in a negative u , which isnot allowed, since by definition u is a positive amplitude.[19] L.-M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, In-separability Criterion for Continuous Variable Systems,Phys. Rev. Lett. , 2722 (2000).[20] R. Simon, Peres-Horodecki Separability Criterion forContinuous Variable Systems, Phys. Rev. Lett. , 2726(2000).[21] J. Heebner, R. Grover, and T. Ibrahim, Optical Mi-croresonators: Theory, Fabrication, and Applications (Springer New York, 2008).[22] S. Shi, Y. Wang, W. Yang, Y. Zheng, and K. Peng, De-tection and perfect fitting of 13.2 db squeezed vacuumstates by considering green-light-induced infrared absorp-tion, Opt. Lett. , 5411 (2018). Appendix A: Simplifying Eq. (18)
In this section we simplify the RHS of Eq. (18) inorder to find a condition for when the equality is true.We define the first two terms on the RHS of (18) to be T d ˆ ρ − / th dt ˆ S † ˆ ρ ˆ S ˆ ρ − / th + ˆ ρ − / th ˆ S † ˆ ρ ˆ S d ˆ ρ − / th dt , (A1)and then using Eq. (16) to simplify, we obtain T d ˆ ρ − / th dt ˆ ρ / th = (cid:88) j =1 (cid:18) − ˆ n j ˙ x j x j + ˆ ˙ x j − x j (cid:19) . (A2)We used the two-mode thermal state in Eq. (4) to getthe last line.We define the middle two terms on the RHS of (18) tobe T ρ − / th d ˆ S † dt ˆ ρ ˆ S ˆ ρ − / th + ˆ ρ − / th ˆ S † ˆ ρ d ˆ Sdt ˆ ρ − / th , (A3)and then using Eq. (16) to simplify, we obtain T ρ − / th d ˆ S † dt ˆ S ˆ ρ / th − ˆ ρ / th d ˆ S † dt ˆ S ˆ ρ − / th . (A4)To simplify this expression we will need to take the timederivative of the two-mode squeezing operator in Eq. (2),which is not straightforward, and thus requires us to becareful. We let ˆ S = exp( σ ), where σ = ξ ∗ ( t )ˆ b ˆ b − h.c. and ˆ S † = exp( − σ ), then d ˆ S † dt = ddt (cid:18) − σ + σ − σ
3! + ... (cid:19) = ∞ (cid:88) n =0 ∞ (cid:88) k =0 ( − n + k +1 σ n ˙ σσ k ( n + k + 1)!= − (cid:90) dλ exp( − λσ ) ˙ σ exp( λσ ) exp( − σ ) , (A5)where the integral in the last line can be shown to beequivalent to the sum on the previous line by expandingthe exponential operators in a power series in σ and doingthe integration over λ from 0 to 1. Multiplying Eq. (A5)by ˆ S from the right, we obtain d ˆ S † dt ˆ S = − (cid:90) dλ exp( − λσ ) ˙ σ exp( λσ ) . (A6)Using the well-known Baker-Campbell-Hausdorff formu-lae on the integrand of Eq. (A6), and then integratingover λ in each term in the series, we obtain d ˆ S † dt ˆ S = − ˙ σ + 12! [ σ, ˙ σ ] −
13! [ σ, [ σ, ˙ σ ]] + ... = ∞ (cid:88) n =1 ( − n L ( n ) n ! , (A7)where the first three terms in L ( n ) are defined as L (1) ≡ ˙ σ , L (2) ≡ [ σ, ˙ σ ], and L (3) ≡ [ σ, [ σ, ˙ σ ]]. In general, we canwrite L ( n ) = [ σ, L ( n − ] for n ≥
2. It is straightforwardto show that, L (1) = ˙ u (cid:16) e − iφ ˆ b ˆ b − e iφ ˆ b † ˆ b † (cid:17) − iu ˙ φ (cid:16) e − iφ ˆ b ˆ b + e iφ ˆ b † ˆ b † (cid:17) (A8) L ( n ) = − i ˙ φ n + ˆ n + 1) (2 u ) n , even n ≥ L ( n ) = − i ˙ φ (cid:16) e − iφ ˆ b ˆ b + e iφ ˆ b † ˆ b † (cid:17) (2 u ) n , odd n ≥ . (A10)We then use Eqs. (A8) to (A10) in Eq. (A7) to sim-plify the derivative. The sum over even n convergesto cosh(2 u ) − n converges tosinh(2 u ) − u . Using these results, we put this simplifiedform of Eq. (A7) into Eq. (A4), along with the two-modethermal state, to obtain T − x x √ x x (cid:18) ˙ u ˆ U + 12 sinh(2 u ) ˙ φ ˆ V (cid:19) , (A11)where ˆ U = ˆ b ˆ b exp( − iφ ) + h.c. and ˆ V = − i ˆ b ˆ b exp( − iφ ) + h.c. . We define the last term of Eq.(18) to be T ρ − / th ˆ S † d ˆ ρdt ˆ S ˆ ρ − / th . (A12)2Using Eq. (3) in Eq. (A12) and simplifying by using Eq. (16) gives T − i (cid:126) (cid:16) ˆ ρ − / th ˆ S † ˆ H ˆ S ˆ ρ / th − ˆ ρ / th ˆ S † ˆ H ˆ S ˆ ρ − / th (cid:17) ++ (cid:88) j =1 Γ j ˆ ρ − / th ˆ S † ˆ b j ˆ S ˆ ρ / th ˆ ρ / th ˆ S † ˆ b † j ˆ S ˆ ρ − / th −− (cid:88) j =1 Γ j (cid:16) ˆ ρ − / th ˆ S † ˆ n j ˆ S ˆ ρ / th + ˆ ρ / th ˆ S † ˆ n j ˆ S ˆ ρ − / th (cid:17) (A13)which can be simplified using the well-known Baker-Campbell-Hausdorff formulae to obtain T − x x √ x x (cid:34) ω + ω u ) ˆ V + (cid:32) i (cid:126) (cid:0) E P γ e − iφ cosh u + E ∗ P γ ∗ e iφ sinh u (cid:1) ˆ U + i ˆ V h.c. (cid:33)(cid:35) ++ (cid:20)(cid:18) x x √ x x − (cid:114) x x (cid:19) Γ + (cid:18) x x √ x x − (cid:114) x x (cid:19) Γ (cid:21)
12 sinh(2 u ) ˆ U + (cid:2) Γ ( x −
1) cosh u + (cid:0) x − − (cid:1) Γ sinh u (cid:3) ˆ n + (cid:2) Γ ( x −
1) cosh u + (cid:0) x − − (cid:1) Γ sinh u (cid:3) ˆ n ++ (cid:2) Γ (cid:0) x cosh u − sinh u (cid:1) + Γ (cid:0) x cosh u − sinh u (cid:1)(cid:3) ˆ . (A14)Using expressions for T T